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You have a number of options. The first and most difficult is to modify or replace the existing door assembly. If you want to use your existing steel door, then the door and jamb assembly must be removed and reinstalled high enough so that the door will clear the floor. This might require some modification in the framing above the door, also. Obviously you will need to do trim replacement and painting/staining both inside and outside the frame. Though difficult, the advantage is you can install ANY FLOORING once you make the change!
Another option is to replace the steel door with a wood door, which can then be cut at the bottom. However, if the new gap between the door bottom and the original threshold is too wide for bottom weatherstripping, you may have to cut out the original threshold and install a "generic" threshold slightly raised to accommodate the shorter door.
If you have access to the basement or crawlspace under the floor, you could reinforce the floor with plywood from underneath, or even install a reinforcing post to stiffen it up.
The less labor-intensive solutions are to use different materials for your entryway. Of course, installing fresh new vinyl flooring is one obvious option. Another is to use slate instead of tile. Slate flooring can generally be installed over thinner floor surfaces than ceramic tile because it is much less prone to breakage over typical 3/4" plywood. Another option is tongue-and-groove laminated flooring. It is often used in kitchens and bathrooms because of its durability and is available in a wide range of styles from simulated wood to subtle colors.